Asymptotic Methods in the Theory of Non-linear Oscillations

A₁ A₁(a A₂ according amplitude approximate solutions assume asymptotic B₁ B₂ characteristic equation coefficients construct corresponding cosy d2x dt2 damping degrees of freedom depending determined differential equations domain dx dt examine expression F₁ forced oscillations formulae Fourier series friction fundamental harmonic Hence inequality initial value integral curves interval Lemma limit cycle linear manifold method monofrequency motion natural frequency non-linear oscillating non-resonance obtain order inclusive oscillating system oscillatory partial derivatives pendulum periodic function periodic solution perturbing force phase plane positive relaxation oscillations represented resonance curves right-hand-side of equation roots satisfied second approximation second order singular point sinusoidal small parameter solution of equation stationary oscillations system of equations t₁ T₂ theorem tion u₁ upto velocity w₁ x₁ zero ε²